A152763 Number of divisors of Catalan number A000108(n).

1, 1, 2, 2, 4, 8, 12, 8, 16, 16, 24, 32, 48, 72, 192, 96, 192, 256, 576, 512, 768, 768, 1024, 1152, 1152, 1728, 1536, 1536, 4096, 4096, 5120, 2048, 6144, 12288, 12288, 8192, 12288, 12288, 24576, 24576, 36864, 98304, 131072, 147456, 196608, 196608, 368640

Offset

0,3

Comments

From Jianing Song, Jun 16 2022: (Start)

Conjecture: a(2^k-1) < a(2^k-2) for all k >= 3. Checked up to k = 263. Note that Catalan(2^k-1) is odd and Catalan(2^k-2)/Catalan(2^k-1) = 2^(k-1)/(2^(k+1)-3). Suppose that 2^(k+1)-3 = Product_{i=1..r} (p_i)^(e_i), let r_i be the (p_i)-adic valuation of binomial(2*(2^k-1),2^k-1), then a(2^k-2)/a(2^k-1) = k * Product_{i=1..r} (e_i-r_i+1)/(e_i+1). This seems unlikely to be less than 1. Actually, it seems that a(2^k-2)/a(2^k-1) tends to infinity as n goes to infinity.

Conjecture: a(2^k-1) != a(2^k) for all k. Checked up to k = 265. Note that Catalan(2^k)/Catalan(2^k-1) = 2 * (2^(k+1)-1)/(2^k+1). Suppose that (2^(k+1)-1)/(2^k+1) = Product_{i=1..r} (p_i)^(e_i), let r_i be the (p_i)-adic valuation of binomial(2*(2^k-1),2^k-1), then a(2^k)/a(2^k-1) = 2 * Product_{i=1..r} (e_i+r_i+1)/(e_i+1). This seems unlikely to be equal to 1. Among the numbers k <= 265, the number k for which a(2^k)/a(2^k-1) is closest to 1 is k = 70, where a(2^k)/a(2^k-1) = 104/105. (End)

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Formula

a(n) = A000005(A000108(n)).

Maple

A000108 := proc(n) binomial(2*n, n)/(n+1) ; end: A152763 := proc(n) numtheory[tau](A000108(n)) ; end: for n from 0 to 80 do printf("%d, ", A152763(n)) ; od: # R. J. Mathar, Dec 15 2008

Mathematica

DivisorSigma[0, CatalanNumber@Range[0, 40]] (* Vladimir Reshetnikov, Nov 13 2015 *)

Prog

(PARI) vector(100, n, n--; numdiv(binomial(2*n, n)/(n+1))) \\ Altug Alkan, Nov 13 2015
(PARI) val(n, p) = (n - vecsum(digits(n, p)))/(p-1); \\ p-adic valuation of n!
a(n) = my(r=1); forprime(p=2, 2*n, r*=val(2*n, p)-val(n, p)-val(n+1, p)+1); r \\ Jianing Song, Jun 16 2022

Crossrefs

Cf. A000005, A000108, A152761, A152762.

Sequence in context: A245597 A324039 A019463 * A221666 A086700 A332004

Adjacent sequences: A152760 A152761 A152762 * A152764 A152765 A152766

Keyword

easy,nonn

Author

Omar E. Pol, Dec 14 2008

Extensions

Extended by R. J. Mathar, Dec 15 2008

Status

approved